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Research Papers

On the Identities for Elastostatic Fundamental Solution and Nonuniqueness of the Traction BIE Solution for Multiconnected Domains

[+] Author and Article Information
Y. J. Liu

Mechanical Engineering,
University of Cincinnati,
Cincinnati, OH, 45221-0072
e-mail: Yijun.Liu@uc.edu

Y. Deng

Mechanical Engineering,
Hong Kong University of Science and Technology,
Hong Kong

1Corresponding author.

Manuscript received September 12, 2012; final manuscript received December 27, 2012; accepted manuscript posted February 12, 2013; published online July 12, 2013. Assoc. Editor: Glaucio H. Paulino.

J. Appl. Mech 80(5), 051012 (Jul 12, 2013) (9 pages) Paper No: JAM-12-1447; doi: 10.1115/1.4023640 History: Received September 12, 2012; Revised December 27, 2012; Accepted February 12, 2013

In this paper, the four integral identities satisfied by the fundamental solution for elastostatic problems are reviewed and slightly different forms of the third and fourth identities are presented. Two new identities, namely the fifth and sixth identities, are derived. These integral identities can be used to develop weakly singular and nonsingular forms of the boundary integral equations (BIEs) for elastostatic problems. They can also be employed to show the nonuniqueness of the solution of the traction (hypersingular) BIE for an elastic body on a multiconnected domain. This nonuniqueness is shown in a general setting in this paper. It is shown that the displacement (singular) BIE does not allow any rigid-body displacement terms, while the traction BIE can have arbitrary rigid-body translation and rotation terms, in the BIE solutions on the edge of a hole or surface of a void. Therefore, the displacement solution from the traction BIE is not unique. A remedy to this nonuniqueness solution problem with the traction BIE is proposed by adopting a dual BIE formulation for problems with multiconnected domains. A few numerical examples using the 2D elastostatic boundary element method for domains with holes are presented to demonstrate the uniqueness properties of the displacement, traction and the dual BIE solutions for multiconnected domain problems.

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Figures

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Fig. 1

A multiconnected domain V with outer boundary So and inner boundary SiSo∪Si = S)

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Fig. 2

Reversing the direction of the normal on Si before applying the identities

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Fig. 10

The last several singular values of matrices [T] shown in (a) and [M] shown in (b) for the plate with four holes. The circles, squares, triangles and stars represent the values obtained from meshes with 140, 280, 560 and 2900 elements, respectively.

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Fig. 3

A square plate with one circular hole

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Fig. 4

The last several singular values of the influence matrix [T] and its submatrices. The singular values are labeled consecutively using integer numbers. Circles, squares, triangles and stars represent the values obtained from meshes with 80, 160, 320, and 640 elements respectively. With the mesh of 640 elements, the condition numbers of [T], [Too],[Toi] and [Tii] shown in (a), (b), (c) and (d), respectively are 2.9 × 106, 1.8 × 106, 3.0 × 107 and 3.2.

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Fig. 5

The last several singular values of the influence matrix [M] and its submatrices. The singular values are labeled consecutively using integer numbers. Circles, squares, triangles and stars represent the values obtained from meshes with 80, 160, 320, and 640 elements respectively. With the mesh of 640 elements, the condition numbers of [M], [Moo],[Moi] and [Mii] shown in (a), (b), (c) and (d), respectively are 3.0 × 104, 1.5 × 104, 5.8 × 107 and 7.4 × 103.

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Fig. 6

The shapes of the plate before and after deformation. The circles show the plate before deformation, and the stars and pentagons represent the deformed shapes obtained from the displacement BIE and dual BIE, respectively. The points marked as diamonds and squares illustrate the two deformed shapes obtained with two different meshes from the traction BIE.

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Fig. 7

The effect of β on the performance of the dual BIE formulation: the last ten singular values of [D] at a few selected β values

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Fig. 8

A schematic of a plate with four circular holes

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Fig. 9

The last several singular values of matrices [T] shown in (a) and [M] shown in (b) for the plate with two holes. The circles, squares, triangles and stars represent the values obtained from meshes with 200, 400, 800 and 1252 elements, respectively.

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